Rocket Propulsion:Fundamental Principles of Rocket Propulsion

Rocket propulsion is based on Newton's three laws of motion:

1. Force equals mass times accleration, $F=ma$; or force equals the change of momentum with respect to time, $F=dp/dt$.
2. The principle of inertia: A body in motion (or at rest) tends to remain in motion (or at rest) unless acted on by a force.
3. Every action produces an equal and opposite reaction.

In general, a rocket works by taking a propellant and accelerating it to go into a particular direction. Because of Newton's third law, the rocket will go in the opposite direction of the propellant. The resulting force that is obtained in propelling the rocket is referred to as thrust.

Most rockets accelerate propellent through a thermodynamic process. For most current rockets, a chemical reaction heats the propellant in a combustion chamber. This heating of the propellant increases its pressure. The combustion chamber has an opening through which the propellant escapes.

Another option is the use of an electric ion thruster. In this case, atoms are stripped of an electron, giving them a positive electric charge. An electric field is set up to propel these ions out of the rocket, again producing thrust.

Alternately, nuclear-reactors may be used to power the rocket. The nuclear reactor may be used to heat the propellant thermodynamically or the nuclear power can be converted to electricity to power an electric ion thruster.

At the simplest level, the trade-off between thermodynamic and electric ion propulsion is thrust versus mass efficiency. Thermodynamic rockets are able to produce high levels of thrust, but are inefficient in their use of propellant. Electric ion propulsion conversely uses propellant very efficiently, but is only able to produce small amounts of thrust, much too small to overcome the Earth's gravity when launching a rocket from the Earth into orbit.

Thermodynamic Analysis

The thermodynamic cycle for a liquid rocket booster is a modified Brayton (jet) cycle. A one-dimensional analysis may be performed by asssuming the following ideal steps.

1. Fuel is injected into a combustion chamber isentropically either through use of pressurized fuel tanks or by a high-pressure pump, increasing the pressure to $p_c$ and increasing the enthalpy.
2. Heat is added to the fuel by means of combustion. In an ideal situation, it is assumed that the pressure remains constant during this step, but that the temperature rises. Both enthalpy and entropy increase during this step.
3. The combusted fuel expand isentropically to the exit pressure, $p_e$, as it goes through the nozzle into the atmosphere, which is at pressure $p_0$. Ideally $p_e$ should equal $p_0$. During this process, the enthalpy decreases from $h_c$ to $h_e$.

The thrust produced by a rocket is given by

$T=\dot{m_p}{v}_e+A_{exit}*(p_e-p_0)$,

where $\dot{m_p}$ and $v_e$ are the mass flow rate and exit velocity of the propellant, $A_{exit}$ is the exit area of the nozzle and $p_e$ and $p_0$ are the pressure at the exit point of the nozzle and the atmospheric pressure. The enthalpy represents the internal energy available for work or the potential energy. Thus, the energy change per unit time as the propellant moves from the combustion chamber to the nozzle exit is

$\dot{m_p}(h_c-h_e)=\frac{1}{2}\dot{m_p}{v}_e^2$.

Solving for the propellant velocity yields

$v_e=\sqrt{2(h_c-h_e)}$.

Let us assume that the combustion mixture of the propellants is an ideal gas. The internal energy per unit mass of an ideal gas is given by

$h={\hat{c}}_{V}T$,

producing an equation for the propellant velocity of

$v_e=\sqrt{2{\hat{c}}_V(T_c-T_e)}=\sqrt{2{\hat{c}}_V{T}_c(1-\frac{T_e}{T_c})}$.

When an ideal gas expands isentropically, a change of temperature and pressure such that the following two relations hold

$\frac{p_1}{p}={(1+\frac{\gamma -1}{2}{M}^2)}^{\gamma /(\gamma -1)}$;

$\frac{T_1}{T}=1+\frac{\gamma -1}{2}{M}^2$;

where M represents the Mach number at the location having static pressure p and temperature T. Using these two equations, we can relate the temperature and pressure ratios as

$\frac{T_1}{T}={\left(\frac{p_1}{p}\right)}^{(\gamma -1)/\gamma }$.

Thus, we can rewrite the equation for the propellant velocity as

$v_e=\sqrt{2{\hat{c}}_V(T_c-T_e)}=\sqrt{2{\hat{c}}_V{T}_c[1-{\left(\frac{p_e}{p_c}\right)}^{(\gamma -1)/\gamma }]}$.

The final analysis step in the one-dimensional analysis is the effects of the nozzle. The previous equation demonstrates that making the ratio $p_e/p_c$ as small as possible maximizes the propellant speed, which in turn maximizes the thrust. The nozzle is designed to match the exit pressure as close as possible to the pressure of the atmosphere or the vacuum of space.

A common metric describing the efficiency of fuel use of a rocket (or jet) is the specific impulse, which is defined as

$I_{sp}=\frac{T}{m_p}$.

As an example, the use of $H_2+0_2$ produces a specific impulse of 454 "seconds".